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09_04_09 - STAT 409 1 Let X 1 X 2 X n be a random sample of...

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STAT 409 Examples for 09/04/2009 Fall 2009 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( 29 < < = - otherwise 0 1 0 θ 1 X θ θ ; x x x f 0 < θ < . a) Find the method of moments estimator θ ~ of θ . μ = ( 29 ( 29 ( - - = = 1 0 1 θ X θ θ X E ; dx x x dx x f x = ( 29 1 θ θ 0 1 1 θ 1 θ θ 1 θ 1 0 θ + = + = + x dx x . 1 θ ~ θ ~ X + = . X 1 X θ ~ - = . b) Is θ ~ an unbiased estimator for θ ? Let g ( x ) = x x - 1 . Then θ ~ = g ( X ) , g ( μ ) = θ . Recall Jensen’s Inequality: If g is convex on an open interval I and X is a random variable whose support is contained in I and has finite expectation, then E [ g ( X ) ] g [ E ( X ) ] . If g is strictly convex then the inequality is strict, unless X is a constant random variable.
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g ' ( x ) = ( 29 2 1 1 x - , g " ( x ) = ( 29 3 1 2 x - > 0 for 0 < x < 1. E ( θ ~ ) = E [ g ( X ) ] > g [ E ( X ) ] = g ( μ ) = θ . θ ~ is NOT an unbiased estimator for θ . c) Is θ ~ a consistent estimator for θ ? By WLLN, X P μ = 1 θ θ + . Since g ( x ) = x x - 1 is continuous at μ = 1 θ θ + , θ ~ = g ( X ) P g ( μ ) = θ . θ ~ is a consistent estimator for θ . d) Find the maximum likelihood estimator θ ˆ of θ . Likelihood function: L( θ ) = ( 29 1 θ 1 1 X X θ θ
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